1211.5945 (R. Toenjes et al.)

Spectral and non-spectral relaxation rates in one dimensional
Ornstein-Uhlenbeck processes    [PDF]

R. Toenjes, I. M. Sokolov, E. B. Postnikov

The relaxation of a dissipative system to its equilibrium state often shows a multiexponential pattern with relaxation rates, which are typically considered to be an intrinsic property of the system, independent of the initial condition and given by the spectrum of a Hermitian operator to which the initial Fokker-Planck operator is transformed by a similarity transformation. We show that this is not always the case. Considering the exactly solvable examples of standard and generalized L\'evy Ornstein-Uhlenbeck processes with \alpha-stable initial distributions we show that the relaxation rates belong the spectrum of the corresponding quantum harmonic oscillator Hamiltonian only if the initial distribution belongs to the domain of attraction of the stable distribution defining the noise. Thus, in case of the standard Ornstein-Uhlenbeck process, broad \alpha-stable initial distributions show a different relaxation pattern, and this pattern can persist as a long transient even for truncated stable initial distributions.

View original: http://arxiv.org/abs/1211.5945